Generally speaking, the audio signal is also analogue at the USB-C port. A pure USB-C jack adapter itself has no other function than to tap the analogue audio signal that is present at the USB-C port. USB Type-C therefore provides an analogue audio signal for headphones and mic. Without any DAC at all. Since you write "adapter", what kind is it?

When you connect a headphone amplifier, you can't just say that it's louder or quieter. That depends entirely on the power of the amplifier, which can vary considerably from one headphone amplifier to the next. It also depends on the impedance of your headphones.

To my ears, the WF-1000XM4 are too loud at full volume to listen to music comfortably. Also, these plug-ins cannot be compared to good wired in-ears, as they simply cannot be connected by cable. With the latter, it also depends on how good the headphone amplifier in the smartphone is and what impedance the in-ears have. As a rule, these are low impedance, as they are intended for portable use and people rarely carry a really powerful headphone amplifier around with them.

So your question is not very clear. First you ask about wired headphones connected to the smartphone with a USB-C to jack adapter (there are no electronics in the adapter, see explanation above), then you mention a HIFI DAC headphone amplifier. But that is more than just an adapter.